Algebra Preliminary Exam, Fall 1980
- Suppose that for each prime integer p dividing the order of a
finite group G, there is a subgroup of index p. Prove that G
cannot be a nonabelian simple group.
- A subgroup of a finite group is ``p-local" if it is the
normalizer of some Sylow p-subgroup. Show that the number of
p-local subgroups of a group is congruent to 1 modulo p.
- Characterize (with proof) all finitely generated
-modules with the property that each submodule is a
direct summand. ( denotes the field of rational numbers.)
- Let R and S be local Noetherian integral domains with
maximal ideals M and N respectively. Assume that
and that S is a finitely generated R-module. If there exists a
proper ideal I of R such that and the canonical
image of R/I in S/IS equals S/IS, then prove that R=S.
- Let R be an integral domain. For , define
. Let be a set of prime ideals of R with the property that
if and , then for some
. Prove that . ( denotes the localization at
- If G is a finite group, prove that there exist fields K and
L such that L is a Galois extension of K with Galois group
isomorphic to G.
- Show that for each positive integer n, there is a polynomial
such that for each matrix
A with complex entries,
( denotes the field of complex numbers and denotes
the trace of A.)
Thu Aug 1 09:44:57 EDT 1996